So assuming the man’s first marriage is valid, the woman is living with a separated man with whom she has children.
The Holy See did make a statement on this situation once. There is the possibility of taking a brother-sister vow of no sexual relations, and also they must sleep and behave in such a way as to not cause sexual temptations to each other, and they must confess their sins, and receive absolution. Then they may remain together to raise the children. If they receive communion it must not be a cause of scandal, so they might have to go to a church where they are not known.
“However, the Church reaffirms her practice, which is based upon Sacred Scripture, of not admitting to Eucharistic Communion divorced persons who have remarried. They are unable to be admitted thereto from the fact that their state and condition of life objectively contradict that union of love between Christ and the Church which is signified and effected by the Eucharist. Besides this, there is another special pastoral reason: if these people were admitted to the Eucharist, the faithful would be led into error and confusion regarding the Church’s teaching about the indissolubility of marriage.”
“Reconciliation in the sacrament of Penance which would open the way to the Eucharist, can only be granted to those who, repenting of having broken the sign of the Covenant and of fidelity to Christ, are sincerely ready to undertake a way of life that is no longer in contradiction to the indissolubility of marriage. This means, in practice, that when, for serious reasons, such as for example the children’s upbringing, a man and a woman cannot satisfy the obligation to separate, they “take on themselves the duty to live in complete continence, that is, by abstinence from the acts proper to married couples.”(180)”
180 Pope John Paul II, AAS 72 (1980), 1082
CDF Divorced and Remarried: